| L(s) = 1 | + (−0.783 + 2.26i)2-s + (0.888 − 0.458i)3-s + (−2.93 − 2.31i)4-s + (−2.65 + 0.253i)5-s + (0.340 + 2.37i)6-s + (−1.79 − 1.94i)7-s + (3.50 − 2.25i)8-s + (0.580 − 0.814i)9-s + (1.50 − 6.21i)10-s + (0.421 + 1.21i)11-s + (−3.67 − 0.707i)12-s + (5.53 − 1.62i)13-s + (5.80 − 2.55i)14-s + (−2.24 + 1.44i)15-s + (0.589 + 2.42i)16-s + (−1.01 − 0.404i)17-s + ⋯ |
| L(s) = 1 | + (−0.554 + 1.60i)2-s + (0.513 − 0.264i)3-s + (−1.46 − 1.15i)4-s + (−1.18 + 0.113i)5-s + (0.139 + 0.968i)6-s + (−0.679 − 0.733i)7-s + (1.23 − 0.796i)8-s + (0.193 − 0.271i)9-s + (0.476 − 1.96i)10-s + (0.127 + 0.367i)11-s + (−1.05 − 0.204i)12-s + (1.53 − 0.450i)13-s + (1.55 − 0.681i)14-s + (−0.579 + 0.372i)15-s + (0.147 + 0.607i)16-s + (−0.245 − 0.0982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.770163 + 0.118059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.770163 + 0.118059i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.888 + 0.458i)T \) |
| 7 | \( 1 + (1.79 + 1.94i)T \) |
| 23 | \( 1 + (-4.79 + 0.176i)T \) |
| good | 2 | \( 1 + (0.783 - 2.26i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (2.65 - 0.253i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.421 - 1.21i)T + (-8.64 + 6.79i)T^{2} \) |
| 13 | \( 1 + (-5.53 + 1.62i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (1.01 + 0.404i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-3.52 + 1.41i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.592 + 4.12i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (0.319 + 6.69i)T + (-30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 1.82i)T + (-12.1 - 34.9i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 6.17i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (3.63 + 2.33i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (6.24 + 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.16 - 1.11i)T + (2.52 - 52.9i)T^{2} \) |
| 59 | \( 1 + (1.58 - 6.54i)T + (-52.4 - 27.0i)T^{2} \) |
| 61 | \( 1 + (3.22 + 1.66i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (7.89 - 1.52i)T + (62.2 - 24.9i)T^{2} \) |
| 71 | \( 1 + (-2.78 - 3.20i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.271 + 0.213i)T + (17.2 + 70.9i)T^{2} \) |
| 79 | \( 1 + (5.88 + 5.60i)T + (3.75 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-4.72 - 10.3i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.0581 + 1.21i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (2.45 - 5.38i)T + (-63.5 - 73.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.85338722417835738860958103331, −9.713991858646623674682690810422, −8.892154181420358260254893919772, −8.076253784025815909407927748894, −7.39220243100157721689936729784, −6.81193118593688600067340548593, −5.78726761202623129430488918125, −4.31397889526997184154279208151, −3.37432436197129706606387107036, −0.61839426163179161614038397114,
1.36556362716887420235499100099, 3.17160955554415325679306806931, 3.42444598736380854854699709128, 4.64661999340650302668701018926, 6.36197521233476169743639635686, 7.84466146345122791699219881392, 8.749619633873024486641667818623, 9.073209543048557059893106143752, 10.10455075418886731052941425655, 11.21711850977946402720067235532
